The Paris agreement does not have any legal power anywhere. It is and always has been a framework for voluntary cooperation between nations to reduce harmful climate change. The only thing required of participant nations was for them to submit a plan, called a "Nationally Determined Contribution" or NDC, for what they thought they could do over the next few decades to reduce greenhouse gas emissions and to make a statement that they would give their best efforts to meet the targets in their NDC. That's it. Actually meeting the NDC targets? Voluntary. Contributions to the Green Climate Fund? Also voluntary. Thus, whether the agreement ever got ratified in the Senate was irrelevant because the agreement didn't actually obligate the US to do anything.
Trump leaving the agreement isn't really that big a deal, at least in practical terms. For one thing, a subsequent President could just as easily rejoin the agreement, and four years' delay isn't that long in the grand scheme of emissions mitigation. For another thing, much of the US NDC was based on observations of ongoing trends in the US energy markets. There is a lot of inertia in those markets, and it's really not clear that there is anything Trump can do to stop the shift from coal to gas, or the falling prices of renewables. If large states like California decide to embark on their own mitigation initiatives on top of that, then the US might not stray too far from its NDC projections, despite withdrawing from the agreement.
In political terms, the effect of Trump's announcement is to communicate the President's attitudes toward the whole subject of climate change. Predictably, his base loves it, and his opposition hates it. It's hard to imagine that the announcement really changed anybody's mind about him. He's made his opinions on the subject pretty clear already. Probably the best thing the announcement can do, from Trump's point of view, is to supplant other issues in the news that he'd rather not talk about. In this respect, the announcement has been reasonably effective.
Update: After exchanging some comments from the author of the question, it seems that the intent of the question was to ask about the role of Senate ratification in an arrangement like the Paris agreement. The short answer is that Senate action or lack thereof is mostly irrelevant to the status of the Paris agreement. For more detail, read on.
The first thing to understand is that US law actually recognizes three types of international accords:
- Treaties require ratification by 2/3 of the Senate. They have some special properties that other agreements do not have, but these nuances are frankly beyond my expertise, so I'll leave them for others to elaborate.
- Congressional-Executive Agreements require the signature of the President and approval by a majority of both houses of Congress. They have more or less the same scope and authority as ordinary legislation.
- Sole Executive Agreements require only the signature of the President. They may only cover matters within the scope of the Executive Branch's Constitutional authority.
One source of confusion is that all of these might be referred to as "treaties" in the context of international relations, but only the first counts as a "treaty" in the context of Article II of the US Constitution.
Many people will be surprised to learn that Article II treaties are just a small part of American foreign policy. According to the Wikipedia link above (which itself draws on Treaties and other International Agreements: the Role of the United States Senate, a report published by the Congressional Research Service):
Between 1946 and 1999, the United States completed nearly 16,000 international agreements. Only 912 of those agreements were treaties, submitted to the Senate for approval as outlined in Article II of the United States Constitution. Since the Franklin Roosevelt presidency, only 6% of international accords have been completed as Article II treaties. Most of these executive agreements consist of congressional-executive agreements.
So, where does that leave the Paris agreement? Because the Paris agreement does not make any binding demands on the nations that agree to it, it is trivially within the President's authority. Presidents do not require Congressional approval to announce aspirational goals. Actually attaining those goals might require help from Congress, or it might not, since the US NDC was structured to fit within the regulatory authority the Executive Branch thought it already had. However, these concerns are not particularly relevant to the Paris agreement because it doesn't actually require signatory nations to meet their NDC goals; it only requires them to state those goals, and that is surely within the Executive's authority.